Question

Why is it important to understand the properties of a theoretical distribution of means of samples of size n when in practice you will only select a single such sample? Why empirical mean from a given sample does not necessarily coincide with theoretical mean?

Answer 1

In practice we indeed take only one sample. But to be able to understand how accurate that sample result may be, we need to know the theoretical distribution of sample means. Since many samples of size n can be drawn from the population. If we calculate means of these samples, we may find different means for different samples. Then we can never tell which sample value is correct or which is wrong. Or can we be sure as which sample gives the true value . Certainly no. However we get another distribution of sample means for all these samples. That distribution depends on true population mean and variances. So in order to make meaningful statement about population means, which is the purpose of sampling we need to know the theoretical distribution of sample mean,knowledge of which helps us find confidence interval based on a single sample observed.

Why empirical mean from a sample won't necessarily coincide with theoretical mean is simply understood by the fact that Theoretical mean is the mean of individual sample means. Secondly a sample can never be perfect representation of population and is bound to have sampling fluctuation.